For a closed system, Hermitian Hamiltonian can dominate the unitary evolution of the system. The eigenenergies are real for obeying the conservation of energy. In 1998, Bender et al. [1]} first found that parity-time (PT) symmetrical non-Hermitian system can have real eigenenergies. Then it pushed the theoretical development of complex extension of quantum mechanics. However, for the PT symmetric quantum mechanics theory of closed system, there is still a lack of self-consistent theory [2]} and experimental verification. For open system, non-Hermitian Hamiltonian can be effectively used to describe the process of evolution.
